The need to characterize the lower-dimensional sets on which the (Euclidean) Riesz transform and other singular integrals are bounded in \(L_2\)  led to the development of uniform rectifiability in Euclidean spaces. In the Heisenberg group, intrinsic Lipschitz graphs have been used to study rectifiability [1]}, [2]} and quantitative rectifiability [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, and, although not explicitly stated, it has been anticipated that intrinsic Lipschitz graphs should be the building blocks of uniformly rectifiable sets. Theorem REF  suggests that in \(\mathbb {H}\) , notions of uniform rectifiability based on intrinsic Lipschitz graphs and notions of uniform rectifiability based on singular integrals may diverge, and points to deep differences between the theory of uniform rectifiability in \(\mathbb {H}\)  and its Euclidean counterpart.
